Pirsig 'applies' his theory quite a bit to twentieth century history, presumably so the reader may have opportunity to absorb by inference where explanation was insufficient. He makes some very insightful remarks, I think, but also flubs quite a bit.
In Zen, he discusses at great length his belief that the social troubles of America are largely due to a widespread perception of a loss of 'quality' in life, which often leads to a phobia of technology and a desire for a simpler lifestyle. He makes a great deal of cutting observations about the tendency of science to over-promise and under-deliver in the way of providing a coherent and stable view of causation in reality -- failing at exactly the thing it is supposed to provide.
The more science progresses, the more that had been established science is overturned, the more complex the explanations become, and the less reality tends to make sense. Think of the discovery of quantum mechanics in a society which had finally just grown used to classical physics. He thinks (and I agree) that this flows largely from the nature of formation of hypotheses, which tend to multiply themselves and lead to ever more possibilities to test as more information is uncovered. The reality revealed becomes ever more detailed and less straightforward as time passes.
The lack of a concept of value within science, which stresses as it does objectivity, further tends to numb people who have come to see science as the guiding principle of social direction. Pirsig has a profound grasp of 'what science is and how it works,' and sees extremely clearly that it is being put to use for the wrong purposes. In fact, I think Pirsig is at his best on this subject. Every graduate student of a scientific discipline should read Zen before considering entering this field.
Probably the insight which 'makes the book' in Lila is his observation that modern American culture is largely derived from Indian (Native American) culture (!). He observes that, compared to Europeans, Americans typically value greater simplicity in language, casualness in dress and demeanor, and relaxed and straightforward attitudes toward expression. They are also given to bluntness in expression in confrontational situations and a tendency towards explosive violence when things come to blows. The ideal of the American cowboy is really only a disguised set of Indian attitudes and values. His sympathy towards Indian attitudes and disdain for 'involuted' European sophistication is evident in his writing style, as I mentioned before.
On other subjects, though, his ignorance of history shows, and his stumbles here I think demonstrate many weaknesses of his philosophy. In trying to show how the conflicts of WWII can be understood in terms of his metaphysics, he asserts that Hitler hated the communists because Nazism was an attempt of social value-patterns to control intellectual values, while communism was an attempt to order social patterns according to intellectual values -- his idea of the proper ordering of things, even if in fact socialism was a rather degenerate and dysfunctional intellectual value pattern and something else would have been better.
In fact, Hitler was a socialist and a communist, who saw that the internationalist component of that ideology would not appeal to Germans, so he invented his own nationalist strain in his striving for power. Other than that, Nazism and communism were practically the same thing -- two very similar, rival ideologies. It was a pragmatic decision. There was no hate involved. Further, if he thinks there was no intellectual component to Nazism, or that communism made no social attempts to control intellectual inquiry as the Nazis did, he is profoundly naïve. It is not nearly so simple as he wants to make it. Or complicated, however one chooses to look at it.
He makes a similar mistake in comparing free-markets with socialism. He claims that socialism is an attempt by intellectual values to order social values, while the free-market is purely an agglomeration of custom (social values) with no real intellectual component. I suppose he has never heard of John Locke, Adam Smith, or any of a hundred other individualist philosophers, market theorists, and economists. In general, I cannot think of a social order which has no such intellectual component -- not even the Victorians, whom he loves to pick on for this sin. It is true that the custom-order may grow up spontaneously and organically, but it is almost always immediately followed by intellectual apologists whose theories begin to mold its development by influencing the social climate with their ideas. Which one came first quickly becomes irrelevant.
Humans are rationalizing creatures. I can't see how they would not do so. Even 'give it to me because I'm bigger than you' is a reflection on the nature of things, however crude. So, at the end of the day, even if I work within his metaphysics, there is no such simple distinction to be made, I think, as Pirsig wants to make it. As far as I can tell, all social orders have their social value-patterns and their intellectual and ideological value-patterns, and we are back to doing regular old philosophy in determining who is right and who is wrong and what the truth is. There is no clear higher and lower dynamic because no such static order can be said to lack a presence at either level of reality. Those that begin as ideologies rapidly produce social patterns and those that begin as social patterns rapidly produce ideologies to justify them.
Pirsig almost seems to want to produce these simplistic evaluations in order to make his philosophy appear simple to apply, even where it should be rather clear that this cannot be the case. Perhaps this is an extension of his attitudes towards Victorianism -- if it isn't simple and straightforward, it isn't any good. This habit of his makes many appearances, but possibly the worst is his evaluation of the morality of war versus the execution of criminals. In the first instance, he justifies the American Civil War on the grounds that a lower order value -- biological human life -- was sacrificed for higher-order social and intellectual values -- cohesion of the American nation and the principle of equality. But not two paragraphs later, he says that the execution of criminals is wrong because a human being is not merely a biological pattern, but a pattern of ideas, which is superior to a social pattern.
Criminals are intellectual value-patterns, but soldiers are not?